Removable singularities of weak solutions to the navier-stokes equations

Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that ever, y weak solution u with the property that Sup_tε(a,b)|u(t)|_L(D)≤ε0 is necessarily of class C^∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (x_o, t_o) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|^-1) as x→x ₀ uniforlnly in t in some neighbourliood of t_o, then (x_o,t_o) is actually a removable singularity of u.     

Fisher–KPP equation with advection on the half‐line

We consider the Fisher–KPP equation with advection: u_t=u_xx?βu_x+f(u) on the half‐line x∈(0,∞), with no‐flux boundary condition u_x?βu = 0 at x = 0. We study the influence of the advection coefficient ?β on the long time behavior of the solutions. We show that for any compactly supported, nonnegative initial data, (i) when β∈(0,c₀), the solution converges locally uniformly to a strictly increasing positive stationary solution, (ii) when β∈[c₀,∞), the solution converges locally uniformly to 0, here c₀ is the minimal speed of the traveling waves of the classical Fisher–KPP equation. Moreover, (i) when β > 0, the asymptotic positions of the level sets on the right side of the solution are (β + c₀)t + o(t), and (ii) when β≥c₀, the asymptotic positions of the level sets on the left side are (β ? c₀)t + o(t). Copyright © 2016 John Wiley & Sons, Ltd.     

Monochromatic Hamiltonian t‐tight Berge‐cycles in hypergraphs

In any r‐uniform hypergraph for 2 ≤ t ≤ r we define an r‐uniform t‐tight Berge‐cycle of length ?, denoted by C_?^{(r, t)}, as a sequence of distinct vertices v₁, v₂, … , v_?, such that for each set (v_i, v_{i + 1}, … , v_{i + t ? 1}) of t consecutive vertices on the cycle, there is an edge E_i of that contains these t vertices and the edges E_i are all distinct for i, 1 ≤ i ≤ ?, where ? + j ≡ j. For t = 2 we get the classical Berge‐cycle and for t = r we get the so‐called tight cycle. In this note we formulate the following conjecture. For any fixed 2 ≤ c, t ≤ r satisfying c + t ≤ r + 1 and sufficiently large n, if we color the edges of K_n^(r), the complete r‐uniform hypergraph on n vertices, with c colors, then there is a monochromatic Hamiltonian t‐tight Berge‐cycle. We prove some partial results about this conjecture and we show that if true the conjecture is best possible. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 34–44, 2008     

New bounds for the Kuramoto‐Sivashinsky equation

We show that every L‐periodic mean‐zero solution u of the Kuramoto‐Sivashinsky equation is on average o(L) for L ? 1, in the sense that for any T > 0 the space average of | u(t) | is bounded by for any t > T and any L sufficiently large. For this we argue that on large spatial scales, the solution behaves like an entropy solution of the inviscid Burgers equation. The analysis of this non‐standard perturbation of the Burgers equation is based on a “div‐curl” argument. © 2004 Wiley Periodicals, Inc.     

Global existence,asymptotic behaviour,and global non‐existence of solutions for damped non‐linear wave equations of Kirchhoff type in the whole space

We study the global existence, asymptotic behaviour, and global non‐existence (blow‐up) of solutions for the damped non‐linear wave equation of Kirchhoff type in the whole space: u_tt+u_t=(a+b∥∇u∥^2γ)Δu+∣u∣^αu in ℝ^N×ℝ₊ for a, b⩾0, a+b>0, γ⩾1, and α>0, with initial data u(x, 0)=u₀(x) and u_t(x, 0)=u₁(x). Copyright © 2000 John Wiley & Sons, Ltd.     

New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow

In this note, we prove an ?‐regularity theorem for the Ricci flow. Let (Mⁿ,g(t)) with t ? [?T,0] be a Ricci flow, and let H_x0(y,s) be the conjugate heat kernel centered at some point (x₀,0) in the final time slice. By substituting H_x0(?,s) into Perelman's W‐functional, we obtain a monotone quantity W_x0(s) that we refer to as the pointed entropy. This satisfies W_x0(s) ≤ 0, and W_x0(s) = 0 if and only if (Mⁿ,g(t)) is isometric to the trivial flow on Rⁿ. Then our main theorem asserts the following: There exists ? > 0, depending only on T and on lower scalar curvature and μ‐entropy bounds for the initial slice (Mⁿ,g(?T)) such that W_x0(s) ≥ ?? implies |Rm| ≤ r^?2 on P_{? r}(x₀,0), where r² ≡ |s| and P_ρ(x,t) ≡ B_ρ(x,t) × (t?ρ²,t] is our notation for parabolic balls. The main technical challenge of the theorem is to prove an effective Lipschitz bound in x for the s‐average of W_x(s). To accomplish this, we require a new log‐Sobolev inequality. Perelman's work implies that the metric measure spaces (Mⁿ,g(t),dvol_g(t)) satisfy a log‐Sobolev; we show that this is also true for the heat kernel weighted spaces (Mⁿ,g(t),H_x0(?,t)dvol_g(t)). Our log‐Sobolev constants for these weighted spaces are in fact universal and sharp. The weighted log‐Sobolev has other consequences as well, including certain average Gaussian upper bounds on the conjugate heat kernel. © 2014 Wiley Periodicals, Inc.     

Ω‐estimates related to irreducible algebraic integers

We study large values of the remainder term E_K (x) in the asymptotic formula for the number of irreducible integers in an algebraic number field K. We show that E_K (x) = Ω_± (√(x)(log x)) for certain positive constant B_K, improving in that way the previously best known estimate E_K (x) = Ω_± (x^(1/2)‐ε) for every ε > 0, due to A. Perelli and the present author. Assuming that no entire L‐function from the Selberg class vanishes on the vertical line σ = 1, we show that E_K (x) = Ω_± (√(x)(log log x)^{D (K)‐1}(log x)^‐1), supporting a conjecture raised recently by the author. In particular, it follows that the last omega estimate is a consequence of the Selberg Orthonormality Conjecture (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constant‐sign solutions for singular systems of Fredholm integral equations

We consider the system of Fredholm integral equations where T>0 is fixed and the nonlinearities H_i(t, u₁, u₂, …, u_n) can be singular at t=0 and u_j=0 where j∈{1, 2, …, n}. Criteria are offered for the existence of constant‐sign solutions, i.e. θ_iu_i(t)≥0 for t∈[0, 1] and 1≤i≤n, where θ_i∈{1,?1} is fixed. We also include an example to illustrate the usefulness of the results obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

A best approximation for the solution of one‐dimensional variable‐coefficient Burgers' equation

In this article, an iterative method for the approximate solution to one‐dimensional variable‐coefficient Burgers' equation is proposed in the reproducing kernel space W_(3,2). It is proved that the approximation w_n(x,t) converges to the exact solution u(x,t) for any initial function w₀(x,t) ε W_(3,2), and the approximate solution is the best approximation under a complete normal orthogonal system . Moreover the derivatives of w_n(x,t) are also uniformly convergent to the derivatives of u(x,t).© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A note on the behavior of blow-up solutions for one-phase Stefan problems

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Long‐time behavior of a nonlinear viscoelastic beam equation with past history

This paper is concerned with the existence of a global attractor for the nonlinear viscoelastic beam equation with past history memory where g(u_t) is a damping like | u_t | ^ru_t and f(u) is a source term like | u | ^αu ? | u | ^βu, by considering 0 ≤ β < α and r > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

The effect of a surface energy term on the distribution of phases in an elastic medium with a two‐well elastic potential

We consider the problem of minimizing among functions u:?^d?Ω→?^d, u_∣?Ω=0, and measurable subsets E of Ω. Here f_h⁺, f^? denote quadratic potentials defined on Ω¯×{symmetric d×d matrices}, h is the minimum energy of f_h⁺ and ε(u) is the symmetric gradient of the displacement field u. An equilibrium state û, Ê of J(u,E) is called one‐phase if E=?? or E=Ω, two‐phase otherwise. For two‐phase states, σ∣?E∩Ω∣ measures the effect of the separating surface, and we investigate the way in which the distribution of phases is affected by the choice of the parameters h??, σ>0. Additional results concern the smoothness of two‐phase equilibrium states and the behaviour of inf J(u,E) in the limit σ↓0. Moreover, we discuss the case of additional volume force potentials, and extend the previous results to non‐zero boundary values. Copyright © 2002 John Wiley & Sons, Ltd.     

Existence of a weak solution to the Navier–Stokes equation in a general time‐varying domain by the Rothe method

We assume that Ω^t is a domain in ?³, arbitrarily (but continuously) varying for 0?t?T. We impose no conditions on smoothness or shape of Ω^t. We prove the global in time existence of a weak solution of the Navier–Stokes equation with Dirichlet's homogeneous or inhomogeneous boundary condition in Q_{[0, T)} := {( x , t);0?t?T, x ∈Ω^t}. The solution satisfies the energy‐type inequality and is weakly continuous in dependence of time in a certain sense. As particular examples, we consider flows around rotating bodies and around a body striking a rigid wall. Copyright © 2008 John Wiley & Sons, Ltd.     

A stability result for abstract evolution problems

Consider an abstract evolution problem in a Hilbert space H (1) where A(t) is a linear, closed, densely defined operator in H with domain independent of t ≥ 0 and G(t,u) is a nonlinear operator such that ‖G(t,u)‖a(t) ‖u‖^p, p = const > 1, ‖f(t)‖ ≤ b(t). We allow the spectrum of A(t) to be in the right half‐plane Re(λ) < λ₀(t), λ₀(t) > 0, but assume that lim_{t → ∞}λ₀(t) = 0. Under suitable assumptions on a(t) and b(t), the boundedness of ‖u(t)‖ as t → ∞ is proved. If f(t) = 0, the Lyapunov stability of the zero solution to problem (1) with u₀ = 0 is established. For f ≠ 0, sufficient conditions for the Lyapunov stability are given. The novel point in our study of the stability of the solutions to abstract evolution equations is the possibility for the linear operator A(t) to have spectrum in the half‐plane Re(λ) < λ₀(t) with λ₀(t) > 0 and lim_{t → ∞}λ₀(t) = 0 at a suitable rate. The new technique, proposed in the paper, is based on an application of a novel nonlinear differential inequality. Copyright © 2012 John Wiley & Sons, Ltd.     

Second‐order finite‐volume schemes for a non‐linear hyperbolic equation: error estimate

We study second‐order finite‐volume schemes for the non‐linear hyperbolic equation u_t(x, t) + div F(x, t, u(x, t)) = 0 with initial condition u₀. The main result is the error estimate between the approximate solution given by the scheme and the entropy solution. It is based on some stability properties verified by the scheme and on a discrete entropy inequality. If u₀ ∈ L^∞ ∩ BV_loc(ℝ^N), we get an error estimate of order h^1/4, where h defines the size of the mesh. Copyright © 2000 John Wiley & Sons, Ltd.     

<Emphasis Type="Italic">BV</Emphasis> functions and parabolic initial boundary value problems on domains

Given a uniformly elliptic second order operator on a possibly unbounded domain , let (T(t)) _t≥0 be the semigroup generated by in L ¹(Ω), under homogeneous co-normal boundary conditions on ∂Ω. We show that the limit as t → 0 of the L ¹-norm of the spatial gradient D _x T(t)u ₀ tends to the total variation of the initial datum u ₀, and in particular is finite if and only if u ₀ belongs to BV(Ω). This result is true also for weighted BV spaces. A further characterization of BV functions in terms of the short-time behaviour of (T(t)) _t≥0 is also given.